In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. Specifically, if V is a vector space over a field k, then a linear functional is a linear function from V to k. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The set of all linear functionals from V to k, Hom_K(V,k), is itself a k-vector space. This space is called the dual space of V. If V is a topological vector space, the space of continuous linear functionals — the continuous dual is often simply called the dual space. If V is a Banach space then so is its continuous dual. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

is a linear functional from the space of integrable functions to the reals.

Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional. For more information see bra-ket notation. Fig. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

A generalized function is an example of a linear functional. In mathematics, generalized functions are objects generalizing the notion of functions. ...

The reason for the use of the term "functional" instead of the traditional term "function" is to avoid potential confusion when a vector space is a space of functions, which is often the case. Hence, linear functionals are often, in practice, functionals in the traditional sense of functions of functions. In mathematics, the term functional is applied to certain functions. ... In mathematics, the term functional is applied to certain functions. ...

See also

• Discontinuous linear map
• Riesz representation theorem
• Positive linear functional
• Bilinear form